Stress-Strain Relationships

Types of relationships

Solids are full of gaps between particles by nature. When force is exerted, the particles at the point of exertion are forced to displace. In the presence of a force of attraction, the other particles are pulled towards them, filling the gaps nearby.

When force is removed, the attraction force will pull all the displaced particles back to their original position as that position forms the most stable arrangement in solids. When all particles are back to their original position, there will be no deformation left with respect to the original solid. Under this condition, the solid exhibits an elastic behaviour, as demonstrated in Fig. 4.1.

When the force exerted is too strong, it may overcome the attraction force. In this case, the unlinked particles can only be pulled or pushed towards their original position indirectly upon removal of the exerted force. In most situations, they will fail to return to their original position and other particles will often slide over each other due to a unbalanced force of attraction. As a result, deformation with respect to the original solid can be observed even when there is no external force. Under this condition, the solid exhibits plastic behaviour, as shown in Fig. 4.2.

The stress-strain relationship is usually determined through experiment. The result is plotted on a graph of stress against strain. This graph clearly shows the behaviour of the material under different amounts of stresses, and thus enables us to determine its yield strength for engineering purposes. For example, the stress-strain curve of steel is shown in Fig. 4.3.

From Fig. 4.3, point A is identified as the elastic limit and the corresponding stress is known as yield stress, v. Prior to point A, the body behaviour obeys Hooke’s Law, in which the strain is solely dependent on stress. At point A, increase in stress will cause the body to yield (loss of some attraction forces) and exhibit plastic behaviour. Unlike elastic behaviour, plastic behaviour is unpredictable as the particles do not always move in the same way after they lose their attraction force. Nevertheless, the body can still resist higher stress because most of the particles are still intact. Point В is an optimum point where the ultimate stress resistance, ult, is achieved in the midst of continuous loss of attraction force due to increasing strain and stress. Beyond that point, particles in the body literally break apart and fracture occurs at point C.

Modulus of elasticity is defined as the amount of stress required to produce one unit of strain and measures the rigidity of a body when it is subjected to uniaxial tension or compression while exhibiting elastic behaviour. It can be easily obtained from the graph of stress against strain by calculating its gradient under elastic zone (CM).

FIGURE 4.1 Elastic behaviour of a body at the microscopic level.

FIGURE 4.2 Plastic behaviour of a body at microscopic level.

FIGURE 4.3 Stress-strain curve for steel.

By substituting Eqs. (2.1) and (3.1) into Eq. (4.1), the expression for modulus of elasticity is transformed to:

Modulus of elasticity provides insights into the strength of attraction force between particles. This attraction force acts in all directions. Even when the force exerted at one point in one direction, the attraction force will pull and push all particles around it. Therefore, a body is expected to develop stress and deform in all directions even though the force is exerted in one direction only. The ratio of transverse strain to longitudinal strain is known as Poisson’s ratio, as defined in Fig. 4.4.

In the above expression, e,_nms is transverse strain, which defines the deformation normal to the direction of force. On the other hand, ei_ong is longitudinal strain, which defines the deformation along the direction of the applied force.

Shear modulus measures a body’s shear rigidity, which has a similar context as modulus of elasticity. Likewise, a greater value in shear modulus denotes that the body does not deform in the direction of shear stress easily.

where

FIGURE 4.4 Longitudinal and transverse deformation of a body.

г is shear stress;

Y is shear strain.